A simple and testable model for earthquake clustering

Abstract

Abstract. Earthquakes are regarded as the realization of a point process modeled by a generalized Poisson distribution. We assume that the Gutenberg-Richter law describes the magnitude distribution of all the earthquakes in a sample, with a constant b value. We model the occurrence rate density of earthquakes in space and time as the sum of two terms, one representing the independent, or spontaneous, activity and the other representing the activity induced by previous earthquakes. The first term depends only on space and is modeled by a continuous function of the geometrical coordinates, obtained by smoothing the discrete distribution of the past instrumental seismicity. The second term also depends on time, and it is factorized in two terms that depend on the space distance (according to an isotropic normal distribution) and on the time difference (according to the generalized Omori law), respectively, from the past earthquakes. Knowing the expected rate density, the likelihood of any realization of the process (actually represented by an earthquake catalog) can be computed straightforwardly. This algorithm was used in two ways: (1) during the learning phase, for the maximum likelihood estimate of the few free parameters of the model, and (2)

Is Earthquake Triggering Driven by Small Earthquakes?

“…While there is a significant amount of literature on the b-value, very few studies have measured accurately the α exponent in real seismicity data. Many studies use α = b without justification [Kagan and Knopoff, 1987;Reasenberg and Jones, 1989;Davis and Frohlich, 1991;Console and Murru, 2001;Felzer et al, 2002]. In this case, small earthquakes are just as important as large earthquakes for the triggering process.…”

“…The α-exponent is an important parameter of earthquake interaction that is used in many stochastic models of seismicity or prediction algorithms [Kagan and Knopoff, 1987;Kagan, 1991;Reasenberg, 1985;1999;Ogata, 1988;Reasenberg and Jones, 1989;Console and Murru, 2001;Felzer et al, 2002]. This parameter controls the nature of the seismic activity, that is, the relative role of small compared to large earthquakes.…”

A Magnitude Independent Space-Time Earthquake Clustering Algorithm (Mistic)

“…On the other hand, stochastic declustering methods mostly model space-time-magnitude occurrences of earthquake clusters, in the form of a branching point process (e.g. Kagan, 1991;Ogata, 1998;Console and Murru, 2001;Zhuang et al, 2002Zhuang et al, , 2004. These models generally classify seismicity into two components, the background and the clustered seismicity, according to certain branching rules.…”

On the Nature and Dynamics of the Seismogenetic System of South California, Usa: An Analysis Based on Non-Extensive Statistical Physics

“…Such models include the paradigmatic ETAS (Epidemic Type Aftershock Sequence) which essentially expresses a selfexcited conditional Poisson process, (e.g. Ogata, 1988Ogata, , 1998Zhuang et al, 2002;Helmstetter and Sornette, 2003;Touati et al, 2009 andSegou et al, 2013), proxy-ETAS models (Console and Murru, 2001;Console et al, 2003Console et al, , 2010 the PPE model (Proximity to Past Earthquakes; Marzocchi and Lombardi, 2008), the EEPAS model (Each Earthquake is a Precursor According to Scale; Rhoades, 2007) as well as their variants and derivatives. The second point of view proposes that seismicity is an expression of non-equilibrating fractal tectonic grain that continuously evolves toward a stationary critical state with no characteristic spatiotemporal scale (e.g.…”